Numerical treatment of multidimensional stochastic, competitive and evolutionary models

In this chapter, we present a computational study of multi-dimensional stochastic systems of differential equations. Especially competitive and evolutionary models. In these types of models, a certain number of uncertainties are causing side effects as random excitations and interactions. To numerically solve the considered systems, we propose the Itô-Taylor family schemes for multi-dimensional stochastic differential equations. Either for systems driven by one white noise or for systems having more than one source of excitations. A first-order accuracy is ensured in the approximation of double Itô integrals by using a truncation in the Fourier series expansion. To verify the accuracy of the proposed method, we consider a system of two stochastic differential equations with known analytical solution. Numerical results are presented for a reduced prototype system of the prebiotic evolutionary model.